Or use the following short-hand notation to declare the polynomial ring and inject the variables at the same time:

sage: R.<a,b,c,d> = RR[]

Second, you can't mix the symbolic solver with polynomial algebra. To "solve" (what does that even mean since there is no formula for the roots of polynomials of degree > 5) polynomial equations you should rephrase your question in terms of ideals and term orders.

I'm not sure I understand your first question about "binary digits". It is true that if `a` is a symbolic variable (not a polynomial ring variable), then `a==0` is a symbolic expression that's treated like an equation in some ways.